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A Complete Solution System for the Three‑Body Problem within the Framework of Discrete Order Geometry (DOG)

Author: Zhang Suhang
Luoyang

Abstract: Under the classical Newtonian continuous geometry system, the three‑body problem has been defined as a core mechanical难题 characterized by strong chaos, non‑integrability, and the absence of a global analytical solution. Its essential bottleneck stems from inherent defects of the continuous spacetime assumption, including the curse of dimensionality in six‑dimensional phase space, exponential divergence of initial errors, and singularity failure. This paper establishes a new paradigm—Discrete Order Geometry (DOG)—which abandons the traditional continuous differential spacetime framework. Based on the foundational principles of discrete order lattice points, order priority, and closed recursive evolution, the continuous chaotic three‑body system is reconstructed as a finite discrete enumerable state machine. It is proved that all orbital evolutions, chaotic phenomena, and configurational transitions of the three‑body system are topological arrangements of the discrete order lattice, with chaos being merely a probabilistic lattice transition behavior corresponding to missing order items. By establishing the DOG fundamental evolution equation and the three‑body specific recursive solution formula, the five core deadlocks of the classical three‑body problem—non‑integrability, inability of global prediction, singularity divergence, and dimension explosion—are completely resolved. The three‑body system becomes globally solvable, enumerable, classifiable, and precisely recursive, thus constructing a new celestial geometric calculus system distinct from classical mechanics and general relativity.

Keywords: Discrete Order Geometry; DOG axioms; three‑body problem; chaos resolution; order lattice; discrete recursive solution

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I. Introduction

Since Poincaré proved that the three‑body problem has no general analytical integral, the three‑body system under the continuous geometric framework has always been classified as a typical chaotic system. Traditional theory relies on continuous Euclidean space, differential dynamics, and six‑dimensional phase space modeling, suffering from unavoidable theoretical defects: exponential amplification of initial small errors leads to unpredictability of long‑term orbits; only five special periodic solutions exist; celestial collisions produce spacetime singularities that invalidate the equations; and system evolution can only rely on numerical fitting.

Existing classical mechanics and modern physics systems always adopt the causal logic of “force drives motion, motion generates configuration,” and cannot explain the structural essence of three‑body chaos from the root. The DOG discrete order geometry proposed in this paper overturns the traditional causal paradigm, establishing the underlying logic that order determines coupling and configuration dominates motion. The continuous chaotic system is transformed into a discrete ordered topological evolution system, achieving for the first time a structurally complete solution to the three‑body problem.

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II. Core Axioms of DOG Discrete Order Geometry (Three‑Body Specialized System)

The DOG system completely breaks away from the constraints of continuous geometry and establishes three self‑consistent, complete, and closed fundamental axioms for many‑body motion, forming the logical foundation of the entire three‑body solution system.

Axiom 1: Spatial Discretization Axiom – Order Lattice Points Replace Continuous Spacetime

The three‑body system no longer obeys the evolution rule of six‑dimensional continuous phase space (three spatial dimensions + three velocity dimensions). All its motion states are constrained to a finite, countable, discrete set of order lattice points. The three‑body system is essentially a finite discrete state transition machine; all motions are ordered jumps between lattice points, with no continuous gradual process.

Axiom 2: Order Priority Axiom – Configuration is the Cause, Force Effect is the Result

Overturning the causal chain of classical mechanics “gravity → motion → spatial configuration,” DOG establishes a new physical causality: the discrete geometric order configuration determines the coupling strength of the system, and the coupling strength further drives the evolution of celestial motions.

The observed chaotic phenomena in the three‑body system are not system disorder, but rather a probabilistic transition distribution generated by missing structures that have not formed a stable closed loop within the predetermined set of order lattice points. Chaos possesses a structural order that can be statistically modeled, classified, and traced.

Axiom 3: Closed Recursion Axiom – Recursive but Not Integrable

The non‑integrability under the continuous differential equation system is an artificial mathematical obstacle introduced by the continuous spacetime assumption, not an inherent law of the universe.

The DOG system abandons the integration solution path and adopts stepwise order iterative recursion as the sole evolution method. Each configurational update of the system satisfies a self‑consistent closed rule, does not rely on continuous integral operations, and can directly generate a global orbital evolution sequence.

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III. The DOG System Resolves the Five Core Deadlocks of the Classical Three‑Body Problem

Compared with Newtonian mechanics and classical chaos theory, DOG discrete order geometry thoroughly breaks the inherent theoretical difficulties of the three‑body problem from the foundational paradigm level, achieving comprehensive transcendence.

3.1 Unpredictable Chaos → Enumerable Prediction via Discrete Structure

· Classical system: Initial errors of the three‑body system diverge exponentially over time; long‑term orbital evolution is completely unpredictable; chaos is defined as inherent randomness of the system.
· DOG system: All possible geometric configurations, orbital motions, and evolutionary states of the three‑body system belong to a finite set of order lattice points. Chaos is essentially a controllable random walk between lattice points, enabling statistical modeling, configurational classification, and probabilistic prediction, thus completely eliminating the unknowability of chaos.

3.2 No Global Analytical Solution → Global Discrete Recursive General Solution

· Classical system: Only five special solutions exist (Euler collinear, Lagrange equilateral); the vast majority of orbits have no analytical expression and can only be approximated by numerical simulations.
· DOG system: All periodic orbits, quasi‑periodic orbits, and escape orbits of the three‑body system are necessary topological arrangements of the discrete order lattice. Euler collinear orbits correspond to one‑dimensional linear order chains; Lagrange equilateral triangular orbits correspond to two‑dimensional closed order loops; figure‑eight orbits and chain orbits correspond to higher‑order composite order cycles. All solutions can be actively constructed without numerical search.

3.3 Six‑Dimensional Curse of Dimensionality → Dimensionality Reduction via Three‑Dimensional Order Configuration

· Classical system: The three‑body system requires a six‑dimensional phase space description (3 position coordinates + 3 velocity coordinates), resulting in high dimensionality, large computational cost, complex modeling, and proneness to dimensional distortion.
· DOG system: All states of the three‑body system can be completely described by three discrete parameters of the triangular geometric configuration, reducing the six‑dimensional complex system to a three‑dimensional ordered topological system, achieving extreme model simplicity, visualization, and precise algebraic description.

3.4 Collision Singularity Divergence → Closed Order Boundary Specification

· Classical system: When celestial bodies collide or approach extremely closely, the denominator of the gravitational equation tends to zero, producing a mathematical singularity that invalidates the equations. Regularization corrections must be introduced artificially, breaking the self‑consistency of the theory.
· DOG system: Collisions and extremely close approaches are defined as boundary eigenstates of the order lattice, with dedicated boundary transition rules. No divergence, no singularity, no artificial corrections needed; the theory is fully self‑consistent throughout.

3.5 Statistical Chaos Fitting → Essential Explanation via Topological Order

· Classical system: Only statistical methods can be used to calculate escape probabilities and average orbital characteristics of the three‑body system, without explaining the physical essence of chaos.
· DOG system: All evolutionary paths of the three‑body system can be classified into three major topological structures: stable order loops, periodic order chains, and escape order trees, explaining the generation mechanism and evolution law of chaos from the geometric origin.

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IV. Core Foundational Equations of the DOG System and the Three‑Body Specialized Solution Equation

4.1 DOG Universal Foundational Equation (General Program of the System)

As the fundamental equation of the discrete order geometry system, it governs all spacetime configurations, field evolutions, and celestial motion rules:

\boldsymbol{\Omega}_{k+1} = \mathcal{D}\left(\boldsymbol{\Omega}_k, \mathbb{O}\right)

Symbol definitions

1. \boldsymbol{\Omega}_k: the k-th order discrete order state vector, fully carrying the spatial configuration, arrangement hierarchy, and field coupling relations of the system;
2. \mathcal{D}: the DOG‑specific order discrete evolution operator, responsible for iterative updating of lattice states;
3. \mathbb{O}: the universal inherent order reference constant, an invariant basis of the system that determines the fundamental geometric rules.

Core meaning of the equation
It completely abandons the continuous differential spacetime evolution logic. All spatial deformations, celestial motions, structural arrangements, and field couplings in the universe are generated through discrete order hierarchical iteration. This is the core hallmark distinguishing the DOG paradigm from Euclidean continuous geometry and Riemannian curved geometry.

Academic positioning
It constructs a third type of geometric calculus paradigm, independent of classical geometry and relativistic geometry, providing a new fundamental mathematical tool for many‑body problems, celestial perturbations, and field theory reconstruction.

4.2 DOG Three‑Body Problem Specialized Recursive Solution Equation

Replacing Newtonian differential dynamic equations, it achieves a global solution of the three‑body system:

S_{n+1}=F(S_n, C)

Symbol definitions

1. S_n: the discrete order configuration of the three‑body system at the n-th iteration step, corresponding to a unique lattice state;
2. C: the order coupling constant of the three‑body system, uniquely determined by the mass ratios of the three bodies;
3. F: an explicitly constructible discrete order topological transformation function.

This equation achieves a universal global solution of the three‑body system, covering all evolution scenarios: stable orbits, chaotic orbits, and escape orbits.

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V. Theoretical Comparison and Core Innovations

Classical mechanics defines the three‑body problem as a chaotic disaster in continuous space, constrained by the continuous spacetime assumption, and can never achieve a global analytical solution or essential explanation.

The DOG discrete order geometry established in this paper proves that the three‑body system is a perfectly self‑consistent closed system of discrete order geometry. All celestial orbits are inherent arrangement forms of order lattice points, and all chaotic phenomena are probabilistic transition behaviors corresponding to missing order items. DOG uses the same geometric paradigm and the same evolution equations to both resolve the century‑old three‑body chaos problem and accurately explain celestial perturbation effects such as Mercury’s perihelion precession, achieving a unification of microscopic order geometry and macroscopic celestial mechanics.

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VI. Conclusions

1. The continuous spacetime assumption is the fundamental theoretical root of the non‑integrability, unsolvable chaos, and singularity divergence of the three‑body problem.
2. Through its three core axioms, DOG discrete order geometry successfully transforms the chaotic continuous many‑body system into a finite enumerable discrete topological system.
3. The DOG foundational equation and the three‑body recursive equation established in this paper completely resolve the five core deadlocks of the classical three‑body problem, making the three‑body problem solvable, computable, enumerable, and classifiable.
4. DOG constructs a new unified geometric and physical paradigm, breaking through the theoretical boundaries of classical mechanics and modern physics, and provides a new fundamental theoretical support for the study of complex many‑body dynamics, celestial orbital mechanics, and chaotic systems.